Finite Group Databasebeta

← #10611776

Subgroups — #10611776

Order 640.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4092C211yesmaximal, characteristic, complemented
2D5×#121C211yesmaximal, characteristic
3C5×#448C211yesmaximal, characteristic, complemented
4#65516C211yesmaximal, characteristic, complemented
5#65463C212yesmaximal, complemented
6#65458C212yesmaximal, complemented
7#65322C211yesmaximal, characteristic, complemented
8#65101C212yesmaximal, complemented
9#65098C212yesmaximal, complemented
10#65025C211yesmaximal, characteristic, complemented
11#65015C211yesmaximal, characteristic, complemented
12C22×C5×Q8C2211yescharacteristic, complemented
13C2×D4×D5—21no—
14C2×D4×D5C2211yescharacteristic
15C22×C4×D5C2211yescharacteristic
16C2×C5×SD16C2212yescomplemented
17C5×#121C2211yescharacteristic
18#4092—21no—
19#4092—21no—
20#4092C2212yes—
21#4092C2212yescomplemented
22C2×#688C2212yescomplemented
23#4085—21no—
24#4084C2211yescharacteristic
25#4076C2211yescharacteristic
26C2×#685—21no—
27C2×C4⋊Dic5C2211yescharacteristic, complemented
28C2×#683—21no—
29C2×#676C2212yescomplemented
30#4041C2211yescharacteristic
31D5×C4⋊C4C2211yescharacteristic
32#4030—21no—
33D5×C22⋊C4—21no—
34C5×#103C2212yescomplemented
35C5×C22⋊C8C2211yescharacteristic, complemented
36#3967C2212yescomplemented
37#3964C2212yescomplemented
38C22⋊#168C2211yescharacteristic, complemented
39#3953C2212yescomplemented
40#3952C2212yes—
41#3950C2212yescomplemented
42#3939C2212yescomplemented
43#3193—51nomaximal, Sylow
44C23×D5—21no—
45C23×D5D411yescharacteristic
46C2×C5×Q8—21no—
47C2×C5×Q8—42no—
48C2×C5×Q8C2312yes—
49C2×C5×D4—21no—
50C2×C5×D4C2311yescharacteristic
51C22×C4×C5—21no—
52C22×C4×C5C2311yescharacteristic
53C2×C22⋊D5—21no—
54C2×C22⋊D5—21no—
55C2×C22⋊D5—21no—
56C22×Dic5—21no—
57C22×Dic5D411yescharacteristic
58D4×D5—42no—
59D4×D5—44no—
60C2×D20—21no—
61C2×D20C2311yescharacteristic
62C2×C4×D5—21no—
63C2×C4×D5—42no—
64C2×C4×D5D411yescharacteristic
65C2×C4×D5D411yescharacteristic
66C2×C4×D5C2311yescharacteristic
67C2×C4×D5C2311yescharacteristic
68C5×SD16—42no—
69C2×C5×C8C2312yes—
70C5×C4⋊C4C2311yescharacteristic
71C5×C22⋊C4—21no—
72C22⋊Dic5—21no—
73#688—42no—
74#685—22no—
75#685—22no—
76#685—21no—
77#685—21no—
78C4⋊Dic5—21no—
79C4⋊Dic5C2312yes—
80C4⋊Dic5C2311yescharacteristic
81#683—21no—
82#683—22no—
83#683—22no—
84C2×#168C2312yes—
85#676—42no—
86C2×C22⋊Q8—51no—
87C2×#121—51no—
88#484—52no—
89#480—51no—
90#480—51no—
91#468—52no—
92#468—52no—
93#450—52no—
94#448—51no—
95#448D511yescharacteristic, complemented
96#407—51no—
97C23×C5—21no—
98C23×C5C2×D411yescharacteristic
99C22×D5—22no—
100C22×D5—22no—
101C22×D5—21no—
102C22×D5—41no—
103C22×D5—42no—
104C22×D5—42no—
105C22×D5—42no—
106C22×D5C2×D411yescharacteristic
107C22×D5C2×D411yescharacteristic
108C5×Q8—22no—
109C5×Q8—22no—
110C5×Q8—42no—
111C5×D4—22no—
112C5×D4—42no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—21no—
116C2×C4×C5—42no—
117C2×C4×C5C2×D411yescharacteristic
118C2×C4×C5C2411yescharacteristic
119C22⋊D5—42no—
120C22⋊D5—42no—
121C22⋊D5—42no—
122C2×Dic5—22no—
123C2×Dic5—21no—
124C2×Dic5—41no—
125C2×Dic5—42no—
126C2×Dic5C2×D411yescharacteristic
127C2×Dic5C2×D411yescharacteristic
128D20—22no—
129D20—42no—
130C4×D5—22no—
131C4×D5—22no—
132C4×D5—41no—
133C4×D5—41no—
134C4×D5—42no—
135C5×C8—22no—
136#168—22no—
137C22×Q8C2×D511yescharacteristic, complemented
138C22×D4—51no—
139C22×D4—101no—
140C23×C4—51no—
141C2×SD16—52no—
142C2×SD16C2×D512yescomplemented
143C2×M16—52no—
144C22⋊Q8—52no—
145C22⋊Q8—52no—
146C22⋊Q8—101no—
147C22⋊Q8—101no—
148#121—51no—
149#121—51no—
150#121—51no—
151#121—101no—
152#121—101no—
153#121C2×D511yescharacteristic
154C2×C4⋊C4—51no—
155C2×C4⋊C4—51no—
156C2×C4⋊C4—101no—
157C2×C22⋊C4—101no—
158C2×C22⋊C4—101no—
159#107—52no—
160#107—52no—
161#103—52no—
162#103C2×D512yescomplemented
163#102—52no—
164#102—52no—
165C22⋊C8—51no—
166C22⋊C8—52no—
167C22⋊C8C2×D511yescharacteristic, complemented
168C22×C5—21no—
169C22×C5—21no—
170C22×C5—41no—
171C22×C5—41no—
172C22×C5—41no—
173C22×C5AGL(1,ℤ/8)12yes—
174C22×C5C22×D411yescharacteristic
175C2×D5—21no—
176C2×D5—21no—
177C2×D5—21no—
178C2×D5—21no—
179C2×D5—21no—
180C2×D5—44no—
181C2×D5—41no—
182C2×D5—41no—
183C2×D5—41no—
184C2×D5—84no—
185C2×D5#13712yes—
186C4×C5—21no—
187C4×C5—42no—
188C4×C5—41no—
189C4×C5H(5,𝔽2)12yes—
190Dic5—21no—
191Dic5—21no—
192Dic5—41no—
193Dic5—42no—
194C24—51no—
195C24—101no—
196C2×Q8—21no—
197C2×Q8—42no—
198C2×Q8C22×D512yes—
199C2×D4—21no—
200C2×D4—51no—
201C2×D4—101no—
202C2×D4—101no—
203C2×D4—101no—
204C2×D4—101no—
205C2×D4—202no—
206C2×D4—204no—
207C2×D4C22×D511yescharacteristic
208C22×C4—21no—
209C22×C4—51no—
210C22×C4—51no—
211C22×C4—51no—
212C22×C4—51no—
213C22×C4—51no—
214C22×C4—101no—
215C22×C4—101no—
216C22×C4—202no—
217C22×C4C22×D511yescharacteristic
218SD16—42no—
219SD16—202no—
220M16—202no—
221C2×C8—52no—
222C2×C8C22×D512yes—
223C4⋊C4—52no—
224C4⋊C4—51no—
225C4⋊C4—101no—
226C4⋊C4—102no—
227C4⋊C4—102no—
228C4⋊C4—101no—
229C4⋊C4C22×D511yescharacteristic
230C22⋊C4—21no—
231C22⋊C4—101no—
232C22⋊C4—102no—
233C22⋊C4—102no—
234C22⋊C4—101no—
235C22⋊C4—101no—
236C2×C5—22no—
237C2×C5—81no—
238C2×C5#53411yescharacteristic
239C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
240C2×C5C2×#13711yescharacteristic
241D5—22no—
242D5—41no—
243D5—81no—
244C23—21no—
245C23—51no—
246C23—51no—
247C23—102no—
248C23—102no—
249C23—101no—
250C23—201no—
251C23—202no—
252C23—202no—
253C23—202no—
254C23D4×D511yescharacteristic
255Q8—22no—
256Q8—22no—
257Q8—42no—
258D4—22no—
259D4—42no—
260D4—102no—
261D4—202no—
262D4—202no—
263D4—202no—
264D4—202no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—51no—
270C2×C4—51no—
271C2×C4—102no—
272C2×C4—102no—
273C2×C4—102no—
274C2×C4—101no—
275C2×C4—201no—
276C2×C4—201no—
277C2×C4—202no—
278C2×C4—201no—
279C2×C4—202no—
280C2×C4D4×D511yescharacteristic
281C2×C4C23×D511yescharacteristic
282C8—22no—
283C8—102no—
284C5#319311yescharacteristic, complemented, Sylow
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—52no—
291C22—101no—
292C22—101no—
293C22—101no—
294C22—101no—
295C22—101no—
296C22—204no—
297C22—201no—
298C22—201no—
299C22—201no—
300C22—404no—
301C22#406012yes—
302C22C2×D4×D511yescharacteristic
303C4—21no—
304C4—41no—
305C4—42no—
306C4—101no—
307C4—101no—
308C4—201no—
309C4—202no—
310C4#414412yes—
311C2—22no—
312C2—81no—
313C2—102no—
314C2—201no—
315C2—401no—
316C2#6592311yescharacteristic
317C2C2×#406011yescharacteristic
318C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#448C2
socleC22×C5C22×D4
prime radicalD5×#121C2
hypercenter#448D5
hypocenterC5#3193
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#4092C2semidirect
3. C5×#448C2semidirect
4. #65516C2semidirect
5. #65463C2semidirect
6. #65458C2semidirect
7. #65322C2semidirect
8. #65101C2semidirect
9. #65098C2semidirect
10. #65025C2semidirect
11. #65015C2semidirect
12. C22×C5×Q8C22semidirect
16. C2×C5×SD16C22semidirect
21. #4092C22semidirect
22. C2×#688C22semidirect
27. C2×C4⋊Dic5C22semidirect
29. C2×#676C22semidirect
34. C5×#103C22semidirect
35. C5×C22⋊C8C22semidirect
36. #3967C22semidirect
37. #3964C22semidirect
38. C22⋊#168C22semidirect
39. #3953C22semidirect
41. #3950C22semidirect
42. #3939C22semidirect
95. #448D5semidirect
137. C22×Q8C2×D5semidirect
142. C2×SD16C2×D5semidirect
162. #103C2×D5semidirect
167. C22⋊C8C2×D5semidirect
284. C5#3193semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#40922
D5×#1212
C5×#4482
#655162
#654632
#654582
#653222
#651012
#650982
#650252
#650152
#31935